Understanding Graph: The Pythagorean Theorem

Welcome to the The Known Equation. For those of you who watch The Big Bang Theory, you'll recall Dr. Sheldon Cooper's explanation of the early beginnings of physics to his neighbor, Penny: "What is physics? Physics comes from the ancient Greek word physika. It's at this point that you'll want to start taking notes. Physika means the science of natural things. And it is there in ancient Greece that our story begins." Like physics, the story of the Pythagorean Theorem also begins in ancient Greece. The Pythagorean Theorem describes the relationship between the side measurements of a right triangle. In other words, the adjacent sides to the right angle and the hypotenuse which is the side opposite the right angle. You probably have taught the Pythagorean Theorem many, many ways, and you know that almost all students in high school know it and will be able to recite it right away: a^2 + b^2 = c^2, but if you go a step further and ask them to elaborate on it, to tell you what a, b and c are, or to tell you about the types of triangles to which the theorem could be applied, you will most likely get fewer responses. If you want them to present one practical application or a proof of the theorem, you may even have a harder time finding somebody to actually do it. This example shows memorization of facts alone is meaningless without the appropriate vocabulary, context, and conceptual and historical perspectives. Learning requires deep understanding. Our brain works in a way that remembers much better within a context, and when given small portions of information repeated often, it actually retains it better and longer. When teaching a topic, you have to look at it from all possible angles and use many examples. But why are the Pythagorean Theorem and its converse so important? Why do we need to study them in detail? Here are few of the reasons: The Pythagorean Theorem is the foundation of trigonometry, algebra, and from there, to calculus. It is used in physics, engineering, and construction. For example, it can be used to calculate distances and right angles. The Pythagorean Theorem can help us calculate the distance between two points on a coordinate plane. We can also use its components to calculate the rate of change. Before discussing the Pythagorean Theorem, a teacher must always make sure students have the necessary foundation. This applies to any topic in mathematics. You cannot build on a faulty foundation. Students need to be familiar at least with the following terms and their meanings: Euclidean geometry, Right angle, Right triangle, Hypotenuse, and Legs. In some aspects, mathematics could be regarded as a foreign language; one must know the symbols and their meaning to understand it. In that sense, it is important to know the misconceptions or the false assumptions students have regarding the Pythagorean Theorem. It is important to understand, in essence, the Pythagorean Theorem could involve variables. The concept of a variable is often difficult for concrete thinkers, who need visuals and manipulatives to grasp abstract ideas regarding exchange. It's important for students to understand how potential variables are identified and the relationship which must exist within an equation. Another important aspect is teaching the origins of the theorem and presenting proofs. Proof is a notoriously difficult mathematical concept for students. Proofs are not instruments of justification, but tools of discovery, which help the development of concepts and the refinement of conjectures (Ratner, 2009). In a deductive system such as mathematics, a proof tests a hypothesis only in the sense of validating it once and for all. Why is the study of proof essential as a pedagogical tool? Because it gives the whole picture and helps your brain make the connections between numbers, vocabulary terms, and the history of the discovery. By examining a proof, a student can understand why a certain statement is true. Many mathematics educators argue that explanation should be the primary purpose of proof in the mathematics classroom. The language of proof can be used to communicate and debate ideas with other students and mathematicians. Teaching students how to prove can allow them to independently construct and validate new mathematical knowledge. To support the proofing process, students require basic understanding of logic and the use of language. It's easier for novice thinkers to understand the "door is open" while also saying, "the door is not closed" which is similar to "a=b" and "b=a." When properties become more than mimicry, relationships are established which can be translated into mathematical symbols. The converse of the Pythagorean Theorem is also true, which is another interesting fact (Srinivasan, 2013). As the Pythagorean Theorem states that the sum of the square of two sides of the triangle will equal the square of the longest side, the converse of the Pythagorean Theorem is that if the sum of the square of the two smaller sides equals the square of the largest side, then that triangle is a right triangle. Making more connections and elaborating further, students should learn about the generalized version of the theorem known as Fermat's last theorem – the idea that a certain simple equation has no solutions. In school, we are taught equations are for solving, and they have a right answer if the steps are followed appropriately. But equations were shorthand for the authors, a way of expressing mathematical relationships which might be true or proved. To discover a theorem which has no solution is a novel idea and brings mathematics into a different light.

Most high school students believe Pythagoras invented the Pythagorean Theorem because it bears his name. Pythagoras was an ancient Greek philosopher from circa 560 to circa 480 B.C. who achieved important developments in mathematics, astronomy, and the theory of music. Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figures, and the abstract idea of a proof. Unlike many later Greek mathematicians, who left behind some of the books they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which means that today Pythagoras is a mysterious figure. His followers were sworn to absolute secrecy, and their devotion to their master bordered on cult-like. Pythagoreans followed a strict moral and ethical code, which included vegetarianism because of the belief in the reincarnation of souls. Whether Pythagoras or someone else from his school was the first to discover its proof cannot be validated because no original works are attributed to him (Ratner, 2009). It is believed Pythagoras provided the first proof of what is now labeled the Pythagorean Theorem, even though no evidence exists to support this claim, either. It is very hard to separate fact from legend about the pre-Socratic philosophers, scientists, and mathematicians. But there are sources

(Ratner, 2009) indicating the Pythagorean Theorem was used by the Mesopotamians and the Babylonians long before Pythagoras was born. Pythagorean triples have been discovered in tablets from around 1800 to 1600 B.C. in Egypt and Old Babylonia (Kamel Al-Khaled & Ameen Alawneh). Evidence indicates the first proofs were provided by Babylonian, Mesopotamian, Indian, and Chinese mathematicians who discovered the theorem independently and, in some cases, provided proofs for special cases.

The distance formula is a derivation of the Pythagorean Theorem. When you look at a coordinate plane and have a set of coordinates, you can use the distance formula to solve the distance between the two points. But how does that work? By having students understand the Pythagorean Theorem, they can then see the application when it comes to a coordinate plane. Instead of using a formula to introduce the concept, the Pythagorean Theorem can be introduced through a discovery method using observation with manipulatives. Multiple online plans and apps are available for appropriate audiences to help set the stage for discovering this ancient theorem. Establishing an authentic problem for students to solve helps them think like a mathematician. Moving to an approach using words, accompanied by some sort of visualization, adds another dimension: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). Having students discuss the meaning of this statement helps them to solidify the concept. By including real-life situations or demonstrations, students can relate the necessity of this theorem to what they might encounter each day. This is also an effective place to explore analogies, learning the unknown by comparing it to the known. At this point, using the symbolic language of mathematics is important. By using the discovery exercises, individuals can become familiar with concepts through a concrete exploration which translates into symbols. Here is an example of solving this equation: If the legs of a right triangle are 5 and 12 units respectively, what is the size of the hypotenuse? a = 5, b = 12, c =? a^2 + b^2 = c^2 5^2 + 12^2 = c^2 25 + 144 = c^2 c^2 = 169 c = √169 c = 13 The theorem is of fundamental importance in Euclidean geometry where it serves as a basis for the definition of distance between two points. Anyone who took geometry classes in high school can't fail to remember it long after other mathematical notions have been thoroughly forgotten.

If we know the sides of a triangle, we can always use the Pythagorean Theorem in reverse to determine if we have a right triangle. This is called the "Converse of the Pythagorean Theorem" and it may be stated as: For any three positive numbers a, b, and c such that a^2 + b^2 = c^2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b. A Pythagorean triple by definition "is a set of three integers a, b, and c which form the sides of a right-angled triangle. The smallest Pythagorean triple is the set of numbers 3, 4, 5. Many ancient cultures and civilizations used those simple Pythagorean triples to accurately construct right angles. For example, if a triangle has side lengths of 3, 4, and 5 units, respectively, then the angle opposite the side of 5 units is a right angle. In other words, the converse of the Pythagorean Theorem enables architects and engineers to conclude that an angle is a right angle based on a certain relationship between the sides of a triangle without having to measure it directly. Furthermore, if the square of the hypotenuse is less than the sum of the square of the sides, the triangle is acute. If the square of the hypotenuse is greater than the sum of the square of the sides, it is an obtuse triangle. The Law of Cosines is a generalization of the Pythagorean Theorem. It states that in a triangle, if C is a right angle, then a^2 + b^2 = c^2. The Law of Cosines states that a^2 + b^2 – 2ab cos(C) = c^2. The cosine of angle C would be zero, and the result would be the Pythagorean Theorem. However, if C is not a right angle, then it has lost its special status. You can still use

the Law of Cosines for a triangle to solve for the various sides and angles, as well as the Law of Sines. This might seem like more jargon to those first introduced to the theorem. Yet, when placed within an understandable context, even the most challenging of ideas become clear.

Mathematics and physics are closely related. As a matter of fact, it is physics that has helped mathematics advance as it has. In other words, if physics did not require so much math, the mathematics that we know today would not be as advanced. So it does not come as a surprise that the Pythagorean Theorem pops up often in the field of physics. Much of the mathematical reasoning employed in the typical introductory physics course can be traced to Pythagorean roots planted over 2,000 years ago. The importance of proofs in mathematics cannot be overstated. If you have a conjecture, the only way you can safely be sure of its truth is through a valid mathematical proof. Much of mathematics is devoted to understanding and defining relationships between objects. Physics was created to explain relationships between measurements of physical objects. Therefore, physics is a specific example of mathematics. We might provide this logic problem, but does it work? All mathematics is relational. All physics measures physical relationships. Therefore, all physics is mathematics. More than 300 known proofs are known for the Pythagorean Theorem; one of them was provided from a 12-year-old Einstein. For a better understanding of the concept, it is really important to show at least one proof in class and let students research more. An early 20th century professor, Elisha Scott Loomis, published a book in 1928 called The Pythagorean Proposition, in which he collected, classified, and discussed 370 proofs of the Pythagorean Theorem. The second edition of the book was published in 1940. The book was written to present what is known relative to the Pythagorean Proposition and to set forth certain established facts concerning the algebraic and geometric proofs and the geometric figures pertaining to the Pythagorean Theorem. First, there are but four kinds of demonstrations for the Pythagorean Proposition: Those based upon linear relations. Those based upon comparison of areas (which is implying the Space Concept) – the Geometric Proofs. Those based upon vector operation (implying Direction Concept) – the Quaternionic Proofs. Those based upon mass and velocity (implying the Force Concept) – the Dynamic Proofs. Second, the number of algebraic proofs is limitless. Third, there are only 10 types of geometric figures from which a geometric proof can be deduced. Fourth, the number of geometric proofs is limitless. Fifth, no trigonometric proof is possible. It was believed at that time that there was no trigonometric proof of the Pythagorean Theorem. It was thought that the theorem could not be proved because the identity of sin^2(x) + cos^2(x) =1 could not be used to prove it because it was based on the Pythagorean Theorem from the beginning. However, it has been shown that you can derive this identity independently of the Pythagorean Theorem, and, therefore, it is trigonometric proof of the Pythagorean Theorem. (Zimba, 2009) As we are talking about the Pythagorean Theorem, in the realm of the Known Equation, it is our duty at least this point to mention the unit circle. We all know that the unit circle is actually used to make other parts of mathematics easier and neater. When you work with the angles in all four quadrants of the coordinate plane, the trig ratios for those angles are computed in terms of the values x, y, and r, where r is the radius of the circle that corresponds to the hypotenuse of the right triangle for your angle theta. In the unit circle, the r value is 1, so the radius is 1. The angle that is made from the center of the circle will become theta, and the sin (theta) = y and the cos (theta) = x. Understanding the relationship between the unit circle and the Pythagorean Theorem will help students to apply their knowledge and not just regurgitate it. As we know, the Pythagorean Theorem has the square of sides added to equal the square of the hypotenuse. However, some have tried to generalize to see if this works for other powers, such as 3 or 4. So mathematicians tried to find if the sum of cubes equals another cube. They weren't able to find any. So, they tried to find if the sum of the fourth power equals the fourth power of another. Again, it didn't work. So the generalization consists of the form, an + bn = cn, where n ≥ 3. It is known by the name Fermat's Last Theorem. Proofs of this conjecture existed only for specific exponents. The general solution was provided in 1995 by the English mathematician, Sir Andrew Wiles, 358 years after Fermat formulated it (Ratner, 2009). In conclusion, we can see that a simple formula, a relatively easy topic in mathematics, can be studied in depth, tracing the history of the discovery, going over the known misconceptions, providing proofs, looking into the converse and the generalization, and assuring conceptual understanding.